SI Units and Uncertainties
Unit 1: Measurements
SI Units and Uncertainties
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
SI Unit (Le Système International d’Unités)
Fundamental units
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meter (m)
kilogram (kg)
second (s)
ampere (A)
Kelvin (K)
mole (mol)
candela (cd)
SI Units and Uncertainties
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Derived Units
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Any unit made of 2 or more
fundamental units
m s-1
 m s-2
 Newton (N) = kg m s-2
 Joule (J) = kg m2 s-2
 Watt (W) = kg m2 s-3
 Coulomb (C) = A s
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Estimation with SI Units
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Fundamental Units
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Mass: 1 kg – 2.2lbs / 1 L of H2O /
An avg. person is 50 kg
Length: 1 m - Distance between one’s
hands with outstretched arms
Time: 1 s - Duration of resting heartbeat
Derived Units
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Force: 1 N- weight of an apple
Energy: 1 J- Work lifting an apple off of
the ground
Scientific Notation and Prefixes
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SI prefixes
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Table
1 Gm = 1,000,000,000 m = 1,000,000 km
1 GM = 1 x 109 m
= 1 x 106 km
0.0000000001 s = 1 ?s = ? ms
Uncertainties & Errors
A. Random Errors
1. Readability of an instrument
2. A less than perfect observer
3. Effects of a change in the
surroundings
Can be
reduced by
repeated
readings
B. Systematic Errors
1. A wrongly calibrated instrument
2. An observer is less than perfect
for every measurement in the
same way
Cannot be
reduced by
repeated
readings
Uncertainties & Errors (cont.)
•An experiment is accurate if……
it has a small systematic error
•An experiment is precise if……
it has a small random error
Systematic error
x
Perfect
x
x
x
Random errors
Uncertainties & Errors (cont.)
Accuracy and Precision:
Precise but not
accurate
Accurate but
not precise
Precision– uniformity
Accuracy- conformity
to a standard
Precise and accurate!
Determining the Range of Uncertainty
1) Analogue scales
(rulers,thermometers meters with needles)
±
half of the smallest division
Since the smallest division on the
cylinder is 10 ml, the reading
would be 32 ± 5 ml
50
40
30
20
10
2) Digital scales
±
the smallest division on the readout
If the digital scale reads 5.052g, then
the uncertainty would be ± 0.001g
Absolute Uncertainty- has units of the measurement
Range of Uncertainty (cont.)
3. Significant Figures
If you are given a value without an
uncertainty, assume its uncertainty is
±1 of the last significant figure
Examples:
•The measurement is 14.742 g, the
uncertainty of the measurement is
14.742 ± .001 g
•The measurement is 50ml, the
uncertainty of the measurement is
50 ± 1 ml
Range of Uncertainty (cont.)
4. From repeated measurements
(an average)
Example: A student times a cart going down a
ramp 5 times, and gets these numbers:
2.03 s, 1.89 s, 1.92 s, 2.09 s, 1.96 s
Average: 1.98 s
Find the deviations between the average
value and the largest and smallest values.
Largest: 2.09 - 1.98 = 0.11 s
Smallest: 1.98 - 1.89 = 0.09 s
The average is the best value and the largest
deviation is taken as the uncertainty range:
1.98 ± 0.11 s
Mathematical Representation
of Uncertainty
For calculations, compare the calculated value
without uncertainties (the best value) with the
max and min values with uncertainties in the
calculation.
Example 1:
Find the density of a block of wood if its mass is
15 g ± 1 g and its volume is 5.0 ± 0.3 cm3
Best value
Density =
m
v
= 15 g
5.0 cm
-3
=
3.0
g
cm
3
Mathematical Representation
of Uncertainty
Example 1 (cont.):
Find the density of a block of wood if its mass is
15 g ± 1 g and its volume is 5.0 ± 0.3 cm3
Maximum value:
Density =
m = 16 g
v
4.7 cm3
= 3.40 g cm-3
Minimum value:
Density =
m = 14 g
v
5.3 cm3
= 2.64 g cm-3
Mathematical Representation
of Uncertainty (cont.)
•The uncertainty range of our calculated value
is the largest difference from the best value..
In this case, the density is 3.0 ± 0.4 g cm-3
•The uncertainty in the previous problem could
have been written as a percentage
Dy = 0.4
X 100% = 13%
y
3
In this case, the density is 3.0 g cm-3 ± 13%
Mathematical Representation
of Uncertainty (cont.)
Example #2:
What is the uncertainty of cos q if q = 60o ± 5o?
•Best value of cos q = cos 60o = 0.50
•Max value of cos q = cos 55o = 0.57 Deviates 0.07
•Min value of cos q = cos 65o = 0.42 Deviates 0.08
The largest deviation is taken as the uncertainty range:
In this case, it is 0.50 ± .08 OR 0.50 ± 16%
Mathematical Representation
of Uncertainty: Shortcuts!
Addition and Subtraction:
When 2 or more quantities are added or
subtracted, the overall uncertainty is equal to
the sum of the individual uncertainties.
Dy = Da + Db
Total uncertainty
Uncertainty of
1st quantity
Uncertainty of
2nd quantity
Mathematical Representation
of Uncertainty: Shortcuts! (cont.)
Example for Addition and Subtraction:
•Determine the thickness of a pipe wall if the
external radius is 4.0 ± 0.1 cm and the internal
radius is 3.6 ± 0.1 cm
External radius = 4.0 ± 0.1 cm
Internal radius = 3.6 ± 0.1 cm
Thickness of pipe: 4.0 cm – 3.6 cm = 0.4 cm
Uncertainty = 0.1 cm + 0.1 cm = 0.2 cm
Thickness with uncertainty: 0.4 ± 0.2 cm OR 0.4 cm ± 50%
Mathematical Representation
of Uncertainty: Shortcuts! (cont.)
Multiplication and Division:
The overall uncertainty is approximately equal to
the sum of the percentage (or fractional)
uncertainties of each quantity.
Total percentage/
fractional uncertainty
Dy = Da + Db + Dc
y
a
b
c
Fractional Uncertainties
of each quantity
Denominators
represent best
values
Mathematical Representation
of Uncertainty: Shortcuts! (cont.)
Example for Multiplication and Division:
Using the density example from before (where the
mass was 15 g ± 1 g and its volume is 5.0 ± 0.3 cm3)
Dy = Da + Db =
y
a
b
1 + 0.3 = 0.07 + 0.06 = .13
15
5 ( this means 13%)
13% of 3 g cm-3 is 0.4 g cm-3
The result of this calculation with uncertainty is:
3.0 ± 0.4 g cm-3 or 3.0 g cm-3 ± 13%
Mathematical Representation
of Uncertainty: Shortcuts! (cont.)
For exponential calculations (x2, x3):
Just multiply the exponent by the percentage
(or fractional) uncertainty of the number.
Example:
Cube- each side is 6.0 ± 0.1 cm
Volume = (6 cm)3 = 216 cm3
Percent
= 0.1 x 100 % = 1.7%
uncertainty
6
Uncertainty in value = 3 (1.7%) = ± 5.1% (or 11 cm3)
Therefore the volume is 216 ± 11 cm3
Problems:
If a cube is measured to be 4.0+_
0.1 cm in length along each side.
Calculate the uncertainty in volume.
1.
Answer: 64+_5 Cm
Problem ( IB 2010)
The length of each side of a sugar
cube is measured as 10 mm with an
uncertainty of +_2mm. Which of
the following is the absolute
uncertainty in the volume of the
sugar cube?
a.+_6 mm
c. +_400 mm
b. +_8 mm
d. +_600 mm
Problem:
3. The lengths and width of a
rectangular plates are 50+_0.5
mm and 25+_0.5 mm. Calculate
the best estimate of the
percentage uncertainty in the
calculated area.
a. +_0.02%
c. +_3%
b. +_1 %
d. +_5%